AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, such that:
.
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
are nonzero real numbers such that![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](http://latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](http://latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing .
be a circle and 
a line in its plane. Let 
be a point on , located outside of . Let 
and 
be the tangents from to , where 
and 
are distinct points on . Let 
and 
be two points on . Lines 
and 
intersect line in two points 
and respectively 
. Lines 
and 
intersect the second time circle in points 
and 
. Prove that the tangents from and to are concurrent on line .
and 
be the roots of the equation 
where 
is an odd integer. Let 
is an integer
) = 1 .
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
. Compute the remainder when is divided by 
.
is given in the plane. The incircle of triangle 
has center 
and is tangent to diagonal 
at 
. Let 
be the center of parallelogram . Show that 
and 
are parallel.
is valid while 
is not)
of positive integers such that 
is an integer.
are positive integers such that for every natural number the set of prime factors of 
are same. prove that 
.
square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always two congruent rectangles.
, there exists some positive integer 
that makes the following equation have no integer root 
.
are nonnegative because 
. We aim to show that there must exist a less than 
that cannot be equal to the left-hand side of the given equation. We count the number of possible values of each so that 
.
, assume for the sake of contradiction that 
. When we multiply this by itself 
times we see that 
, which we don't want. to be less than 
. There are possible values for in the range 
. When we take the product of for all such that 
, we get ![\[ \prod\limits_{i=1}^n 2^{2^{n-i}} = 2^{\sum\limits_{i=1}^n 2^{n-i}} = 2^{2^n - 1}\]](http://latex.artofproblemsolving.com/e/4/d/e4d0e5a2f1977c4e12bec6fa560fbe9166f30418.png)
This value is an upper bound for the amount of tuples 
such that the LHS of the given equation is less than . However, this value 
is less than the 
possible values of less than . Thus, by pigeonhole there must exist a less than that cannot be equal to the LHS of the given equation.
Prove that
Where 
is always false. If it turns out this equation can be true for some large